Heat the oven to 350°F. Arrange the chicken in a single layer on a rimmed baking sheet or plate. Season with 2 tsp. of the salt and 1 tsp. of the pepper. (I like to massage the seasoning into the chicken to ensure it’s evenly distributed.) Place the flour in a shallow dish. Dip the chicken in the flour, making sure to coat it on all sides, then return it to the sheet.
Place a large (12- to 14-inch), straight-sided sauté pan or a Dutch oven over high heat for several minutes. When hot, add 1/4 cup of the oil and heat until shimmering. Add the chicken, skin side down (that way the fat “melts” as it cooks and gives you a bit more sizzle in the pan), in a single layer and sauté over high heat without moving it for about 4 minutes, or until browned on the first side. (If the pieces stick, that means they haven’t browned long enough. Let them cook a bit longer. If sticking continues to be a problem, use a sharp-edged spatula to loosen the skin from the pan.) Turn the chicken over and repeat, lowering the heat to medium-high if it gets too dark too quickly. Transfer the browned chicken to a clean baking sheet, trying to leave as much oil as possible in the pan.
Let the pan heat up for a minute over high heat. Add the mushrooms, season with the remaining 1/2 tsp. salt and 1/8 tsp. black pepper, and let them sauté, without stirring, for about 2 minutes. Stir and let the mushrooms sauté undisturbed for another 2 minutes. Stir again and repeat, if necessary, until they’re almost all seared and golden. Remove the mushrooms to the baking sheet with the chicken, leaving as much fat in the pan as possible.
Reduce the heat to medium-high. Add the remaining 3 Tbs. olive oil and then the onions, carrots, and peppers. Sauté until the onions are very soft, 10 to 15 minutes. Add the garlic and cook for 1 minute more, until fragrant.
Deglaze the pan with the wine, stirring to scrape up the browned bits. Add the tomatoes, squeezing them (with clean hands) as you add them to the pan to help them break down faster. Return the chicken and mushrooms to the pan and bring to a simmer. Cover the pan tightly with foil or a lid and place in the oven. Cook
for about 40 minutes, or until the chicken is very tender.
The sauce should be the perfect consistency (like a rich soup rather than pasta sauce), but if it seems at all watery, transfer the chicken to a plate and tent with foil to keep warm. Simmer the sauce, uncovered, over medium-high heat until thickened, about 5 minutes. (How long this takes depends on how thin the sauce is and the dimensions of your pan. Just keep an eye on it so you don’t over-reduce it). Taste and add 1 more tsp. salt if needed. Serve with rice, pasta, or crusty bread.